A winning strategy is a plan that reaches the stated winning condition against every legal reply, not merely against the reply we hope an opponent will choose. In a small mathematical game, a move is useful because of the position it leaves for the other player. Counting, remainders, symmetry and backwards reasoning can turn apparently random choices into an explainable plan.
Place twenty-six counters in a pile. Two players alternate removing one, two or three counters, and whoever takes the last counter wins. Removing the largest allowed number looks attractive, but the stronger opening is to remove two, leaving twenty-four. Whatever the opponent removes next, respond so that their removal and yours total four. The pile then returns to twenty, sixteen, twelve, eight, four and eventually zero on your move.
This guide explains why that strategy works, when it fails and how to rebuild the reasoning after the rules change. It is Primary Mathematics enrichment, not a claim that formal game theory is required in every Primary syllabus. The earlier Working Backwards, Invariants and Remainder Cycles guides provide useful prerequisites.
All activities use counters, numbers or classroom points, without money or betting. They are fully specified finite games with visible states. A mathematically losing position describes what can be forced under the rules; it does not label a child as a poor thinker or predict how an informal game between learners will actually finish.
Use the MOE Primary curriculum page and the learner’s school programme for required scope. NRICH’s Solving Together: Got It! offers related further exploration of target games and rule changes. The games below have their own stated numerical examples, original questions and self-contained proofs.
Specify the game · Winning and losing states · Take one to three · Last counter loses · Target games · Symmetry · Changed move sets · 24 questions · Worked answers · Teaching
1. A strategy belongs to a complete set of rules
Before analysing a game, state the starting position, whose turn it is, the legal moves and the terminal condition. For our first pile game, the complete state is the number of counters remaining together with the player whose turn it is. A move removes one, two or three counters, but never more counters than remain. Taking the last wins immediately.
There are no passes, no hidden counters, no chance events and no simultaneous moves. Both players see the pile. Each move decreases a non-negative whole-number quantity, so the game must end after finitely many moves. That termination fact matters: a plan that merely keeps avoiding defeat forever would not be a winning strategy in a game where draws or endless play are possible.
A rule change can reverse a move’s meaning
If taking the last counter loses rather than wins, a pile of one changes from an immediate win to an immediate loss for the player to move. The later strategic classes change as a consequence. Do not carry the old list of preferred pile sizes into the new game without rechecking the ending.
Similarly, allowing only removals of one, three or four is not the same as allowing every removal from one through four. The missing move two can break a response that an old strategy requires. “The largest allowed move is four” does not completely describe the legal move set.
A state must contain everything needed for future decisions
For one pile with fixed rules, the remaining count is enough apart from turn identity. For two piles, both pile sizes matter. If a rule depends on the previous removal, that previous move also becomes part of the state. Two positions displaying the same counter total may therefore have different strategic possibilities.
This is why a pattern noticed in a score alone may not be a reliable strategy. Ask whether the recorded state includes every condition that affects future legal moves. A simpler record is useful only if it preserves the information needed by the game.
Playing well is not always taking more
A move can leave fewer counters while giving the opponent a forced win. Another move can leave more counters but put the opponent in a position from which every legal move gives you a winning response. The objective is the final rule, not the largest immediate collection of counters.
The examples here provide an inspectable way to test that distinction. A claim such as “take three whenever possible” can be challenged with one pile where that move loses and a smaller removal wins.
2. Work backwards from positions you can settle immediately
Call a position winning for the player to move when at least one legal move guarantees a win against perfect replies. Call it losing when every legal move lets the opponent force a win. The words describe the state under ideal play, not the eventual score when someone makes a mistake.
For the one-to-three removal game with last-counter-wins, a pile of one, two or three is winning: remove everything. A pile of four is losing. Whatever is removed, one, two or three remains, so the opponent can take the rest and win.
The two tests use different quantifiers
To establish a winning position, find one legal move to a losing position for the opponent. To establish a losing position, show that every legal move goes to a winning position for the opponent. A single bad move does not make the original state losing, because another move might work.
This is a direct application of the Logic guide. “There exists a successful move” and “all moves fail against best play” need different evidence. Confusing them can make a partial search look like a complete strategic conclusion.
Small states provide a foundation
Five is winning: remove one and leave four. Six is winning: remove two and leave four. Seven is winning: remove three and leave four. Eight is losing because its possible remaining piles, seven, six and five, are all winning for the next player.
Continue this backwards-style classification by increasing pile size, using results already established for smaller piles. The direction of arithmetic is forward through the numbers, but the reasoning asks what each move leaves at the next stage.
| Counters before your move | State under last-counter-wins | One reason |
|---|---|---|
| 1, 2 or 3 | Winning | Take the whole pile |
| 4 | Losing | Every move leaves an immediate win |
| 5, 6 or 7 | Winning | Leave 4 |
| 8 | Losing | Every move leaves 5, 6 or 7 |
| 9, 10 or 11 | Winning | Leave 8 |
| 12 | Losing | Every move leaves a winning state |
A losing state still requires the opponent to use the opportunity
If you begin at four and remove one, your opponent could make the mistake of taking only one from the remaining three. You may then win. That possibility does not make four a winning state under perfect play. It shows the difference between a state classification and the choices made in one actual game.
A useful classroom question is therefore “Can this player force a win?” rather than “Could this player possibly win?” Many losing positions still permit victory after an opponent’s error.
3. Why multiples of four control the one-to-three game
The losing positive pile sizes are four, eight, twelve, sixteen and every later multiple of four. To justify the whole pattern, do not rely only on the first few table entries. Explain how the legal moves connect the classes.
From a positive multiple of four, removing one, two or three leaves a non-multiple. From any non-multiple, remove its remainder on division by four: one, two or three. That is a legal removal and leaves a multiple of four. Near the end, a pile of one, two or three can be taken completely.
These two facts, together with the ending, explain why the pattern continues. Every legal move out of a multiple gives the opponent a way back to a smaller multiple; every non-multiple has a move into the losing class. The finite decrease ensures that maintaining the strategy reaches the last-counter win rather than merely postponing an ending.
Example A: Starting with twenty-six
Twenty-six leaves remainder two on division by four. Remove two to leave twenty-four. If the opponent removes one, remove three. If the opponent removes two, remove two. If the opponent removes three, remove one. Each complete pair removes four and restores a multiple-of-four pile for the opponent.
The target piles after your moves are twenty-four, twenty, sixteen, twelve, eight, four and zero. At four, the opponent must leave one, two or three, which you take. Every response in the plan is legal, including the final response.
Example B: A tempting wrong opening
At eighteen, removing three leaves fifteen. The opponent can remove three to leave twelve, taking control of the multiple-of-four pattern. The correct strategic opening is to remove two, leaving sixteen. Taking fewer now can secure the last counter later.
A counterexample like this disproves “always take the maximum.” It does not show that taking three is never correct; at nineteen, removing three leaves sixteen and is precisely the winning opening. The right move depends on the state.
Starting on a multiple
At twenty-four, no first move can force a win against an opponent who responds correctly. The second player can complete each pair to four. There is no hidden opening that escapes the classification while all the stated rules remain unchanged.
That is a mathematical result about this game, not a reason to stop exploring. Change the starting number, legal removals or ending rule and ask which part of the proof survives. Rule variation is useful precisely because it tests whether the learner understood the explanation rather than remembered one opening.
The general one-through-m version
If the permitted removals are every whole number from one through m and taking the last wins, the corresponding block size is m + 1. A removal k can be answered by m + 1 − k, which is also permitted. Non-multiples can be moved to multiples by removing their non-zero remainder.
The word every is essential. This proof does not cover a move set with missing values. It also assumes there is only one pile and that no additional rule restricts the responding move. A generalisation should carry its assumptions along with its formula.
4. Changing the final condition changes the strategic pattern
Keep the same one-to-three removal choices, but now taking the last counter loses immediately. With one counter, the player to move is forced to take it and lose. With two, remove one and leave the opponent the last. With three, remove two. With four, remove three. All three of those positions are winning.
At five, every removal leaves two, three or four, each a winning state for the opponent. Thus five is losing. The losing positive counts become one, five, nine, thirteen and so on: numbers leaving remainder one on division by four.
Example C: Starting with eighteen
Remove one to leave seventeen, which is one more than a multiple of four. After the opponent removes k counters, remove four minus k. This returns the opponent to thirteen, nine, five and finally one. The opponent then must take the last counter and lose.
The response sizes look familiar, but the preserved residue has changed. In the last-counter-wins game, you leave multiples of four; here you leave one more than a multiple. Keeping the old targets would give the wrong player the last counter.
Do not apply the response mechanically at the terminal point
Once the opponent faces one counter, the game ends on their required removal. There is no further response to make. A strategy description should include when to stop, not only an arithmetic pattern for middle moves.
When analysing the zero-counter state, be careful about perspective: under this rule the game has already ended because someone took the last and lost. The simplest Primary explanation begins at one counter rather than treating empty piles as ordinary playable positions.
Check every legal reply at a losing state
At five, the legal remaining piles are four, three and two. The opponent can leave one from each. Removing more than three is not allowed, so there is no overlooked move directly to one. The all-replies check establishes losing status without relying on how one practice game happened to unfold.
A learner who says “five is losing because I lost yesterday” has evidence about a trial, not yet a proof. Ask them to list the legal choices and the opponent’s response to each.
5. Counting towards a target is the same structure with a different record
Start with a running total of zero. Players alternate adding one, two, three or four. The player who reaches exactly thirty-one wins, and moves beyond thirty-one are forbidden. Instead of removing counters, the game reduces the distance remaining to the target.
Initially thirty-one remains. A first addition of one leaves thirty to reach, a multiple of five. Then answer an opponent’s addition k with five minus k. The totals after your moves are one, six, eleven, sixteen, twenty-one, twenty-six and thirty-one.
Example D: Explain a middle response
You have just reached eleven. The opponent adds four, reaching fifteen. Add one to reach sixteen. Their move and yours add five in total, maintaining the sequence of target totals under your control. If they had added two, you would add three instead.
The plan is not “always add one.” One is the opening and the response to four, but other opponent moves require other responses. A strategy is a rule for choosing a move from the current situation, not a fixed string of moves regardless of what happens.
Example E: A target that favours the second player
Change the target to thirty while keeping additions one through four and starting from zero. The distance remaining is initially a multiple of five. Every first addition allows the second player to complete the pair to five and then preserve totals five, ten, fifteen, twenty, twenty-five and thirty.
Under perfect play, the first player cannot force a win from zero in this version. The roles change because the target changed relative to the five-step block, even though the legal additions stayed the same.
Exact-target rules must be checked near the end
At total twenty-nine in the thirty-one game, adding two wins immediately. Adding three or four is illegal because it overshoots. A move that is allowed earlier may become unavailable near the target. The proof works because its prescribed winning response always reaches a valid target total without overshooting.
A variant allowing a win on reaching or passing the target should be analysed under its own wording. Do not assume every rule change necessarily changes every position, but do not assume the old proof applies without checking the changed terminal conditions either.
Backwards landmarks are easier than memorised openings
From the target thirty-one, subtract five repeatedly to obtain twenty-six, twenty-one, sixteen, eleven, six and one. These are the totals a player seeks to leave after their turns. The backwards sequence explains the opener one rather than presenting it as a lucky discovery.
This connects target games to ordinary reverse-process problems. The difference is that a game strategy must also provide a response for each choice made by another player between your controlled landmarks.
6. Two equal piles can support a mirror strategy
Consider two piles with six counters each. On a turn, a player removes any positive number from exactly one pile, no more than it contains. Taking the final counter of both piles wins. The second player can mirror each first-player move in the other pile.
If the first player removes two from the left, the second removes two from the right. If the first player next removes one from the right, the second removes one from the left. After each response, the piles are equal again. The second player’s move is always legal because the other pile had the same size before the pair of moves.
Why symmetry gives the final move
When both piles are equal and non-empty before the first player’s turn, that player cannot empty both in one move because only one pile may be touched. The second can mirror the removal. Eventually the first empties one of two equal piles, and the second empties the other, taking the last counter.
The total decreases on every move, so this ending cannot be postponed indefinitely. The proof includes legality, preservation of equality and the terminal winner. Merely noticing that the drawing is symmetric would not establish all three.
Example F: Unequal piles
Start with piles of nine and six. The first player removes three from the larger pile, making six and six for the opponent. The first player then takes the mirroring role after every opponent move. The initial equalisation converts an unequal start into the same strategic structure.
This proof is specific to two piles and the stated ability to remove any positive number from one pile. If removals were limited to one or two, reducing nine to six in one move would be illegal. A familiar-looking diagram does not override a changed move rule.
More piles require another argument
Do not extend “make two piles equal” into a universal recipe for every multi-pile game. A third pile can create moves not covered by the two-pile mirror proof. Our two-pile result is useful and complete within its scope; claiming a larger theorem without checking it would weaken the explanation.
The same discipline applies in geometry and probability. A successful argument should say exactly which conditions make it work, especially when a tempting generalisation changes the state space.
7. A non-consecutive move set needs a fresh state table
Now use one pile, allow only removals of one, three or four, and let the last-counter taker win. Two is no longer an allowed removal. The earlier “largest move plus one” shortcut would suggest blocks of five, but its response proof requires missing complements and therefore does not apply.
Settle the smallest states. One is winning by taking one. Two is losing: only removing one is legal, leaving one for the opponent. Three and four are winning by taking the entire pile. Five is winning by removing three and leaving two. Thus five is not a losing multiple-of-five state.
Example G: A losing state at seven
From seven, the legal remaining piles are six, four and three. Six is winning because removing four leaves two; four and three are immediate wins. All legal successors are winning for the next player, so seven is losing.
Eight is winning by removing one to leave seven. Nine is losing because its successors eight, six and five are all winning. These conclusions are obtained by checking the actual legal successors, not by importing a pattern from a different game.
| Pile | Status for removals {1, 3, 4}, last wins | A valid reason |
|---|---|---|
| 1 | Winning | Remove 1 |
| 2 | Losing | Only move leaves 1 |
| 3, 4 | Winning | Remove the whole pile |
| 5 | Winning | Remove 3, leaving 2 |
| 6 | Winning | Remove 4, leaving 2 |
| 7 | Losing | Successors 6, 4, 3 are winning |
| 8 | Winning | Remove 1, leaving 7 |
| 9 | Losing | Successors 8, 6, 5 are winning |
| 10, 11, 12 | Winning | Leave 9 or 7 with an allowed move |
A table supports the states it actually covers
The displayed table classifies the stated small range. It may suggest a repeating pattern further on, but a claim about all pile sizes needs an argument that the pattern continues. Extending a table to a hundred entries is stronger computational evidence, not automatically a proof for every integer.
For this guide, the one-to-three game includes a general remainder proof. The non-consecutive game is a lesson in rebuilding the finite analysis after the rules change. Keep those two kinds of evidence distinct.
A strategy needs a reply to every legal response
Writing “I leave seven, then the opponent takes one” checks one branch. At seven the opponent might take three or four. To prove that leaving seven is safe for you, classify all three remaining positions and provide an appropriate winning continuation after each.
A small tree or table is enough for these examples. There is no need to pretend a long computation or autonomous agent is required. The educational value comes from making the alternatives inspectable.
8. Practice: 24 questions
Keep the worked answers covered. For Questions 1–8, one pile is used; players alternate removing one, two or three, never more than remain, and taking the last wins. Winning and losing refer to the player whose turn it is against perfect replies. Later groups explicitly change the rules.
Questions 1–8: Last counter wins, remove one to three
1. Three counters remain on your turn. What move wins immediately?
2. Four counters remain. Can the player to move force a win? Check every legal move.
3. Seven counters remain. Find a winning move and explain the position left.
4. Twelve counters remain. Classify the position and justify it using the repeating structure.
5. Eighteen counters remain. Find the winning first removal.
6. Twenty-one counters remain. Give the first removal and the sequence of pile sizes you aim to leave after your turns.
7. You left sixteen counters. Your opponent removes two. What should you remove next to preserve the strategy?
8. At eighteen counters, a learner removes three because it is the largest move. Explain how the opponent can take control and identify the better opening.
Questions 9–12: Last counter loses, still remove one to three
9. One counter remains. Classify the position for the player to move.
10. Four counters remain. Find a winning removal.
11. Five counters remain. Check every legal move and classify the position.
12. Eighteen counters remain. Find a winning opening and the residue class you aim to leave for the opponent.
Questions 13–16: Add one to four, reach the exact target
13. Start at total zero, add one to four per turn, and win by reaching exactly thirty-one. What should the first player add, and which totals should follow after their later turns?
14. In Question 13, you leave total eleven and the opponent adds four. What is your response?
15. Change the target to thirty, with all other rules unchanged. Which player has a forced win from zero under perfect play?
16. In the exact-thirty-one game, the current total is twenty-nine on your turn. Which addition wins immediately, and which otherwise familiar additions are now illegal?
Questions 17–24: Symmetry, changed moves and evidence
17. Two piles contain five counters each. A move removes any positive number from one pile only; taking the last wins. Explain a second-player strategy.
18. Under the same two-pile rules, start with eight and five. Find a winning first move and continuation.
19. Why is the mirror response always legal when two equal piles are presented to the opponent?
20. Under the two-pile rules, piles are three and three. The first player removes all three from one pile. What should the second do, and who wins?
21. Return to one pile, allow only removals {1, 3, 4}, and let the last taker win. Five remain. Find a winning move and explain why taking four is worse.
22. Under the rules of Question 21, seven remain. Classify the position by checking all legal successors.
23. A player wins five practice games using one method. Does this prove a winning strategy against every legal reply? What evidence is still needed?
24. A table correctly classifies the {1, 3, 4} game for piles up to thirty. Does that alone prove a proposed pattern for all pile sizes? Distinguish the verified range from the general claim.
9. Worked answers and all-replies checks
Answers 1–8
1. Remove all three. Three is an allowed removal and takes the last counter, satisfying the winning condition immediately. There is no need to leave a strategic smaller pile when the game can already be won. Always check the terminal rule before applying a middle-game pattern.
2. No; it is losing under perfect replies. Removing one leaves three, removing two leaves two, and removing three leaves one. In every case the opponent can take the entire remainder and win. All legal first moves have been checked, not merely one unsuccessful choice.
3. Remove three, leaving four. Four is losing for the next player under the all-replies argument in Question 2. Whatever the opponent removes from four, take the remainder. This supplies a legal winning continuation for each reply rather than assuming a particular opponent move.
4. Losing. Twelve is a multiple of four. Every permitted removal leaves nine, ten or eleven, each of which can be reduced to eight. The opponent can then keep returning smaller multiples of four. The initial player has no move directly into the losing class under these rules.
5. Remove two, leaving sixteen. Eighteen has remainder two on division by four. Thereafter answer a removal k with four minus k. The opponent keeps receiving multiples of four, eventually four itself, from which any legal move lets you take the last counter.
6. Remove one. Aim to leave twenty, sixteen, twelve, eight, four and zero after your turns. Each opponent-response pair removes four. At four, the opponent leaves one, two or three; your legal final response takes the rest. The target sequence follows from the rules and termination.
7. Remove two. The opponent changed sixteen to fourteen. Removing two reaches twelve, the next controlled multiple of four. Their move and yours total four. Taking three instead would leave eleven and hand the opponent a route back to the strategic multiples.
8. The opponent removes three from fifteen to leave twelve. They can then complete each pair to four and force the last-counter win. The better opening from eighteen is two, leaving sixteen. This counterexample refutes “always take the most,” without claiming that taking three is never correct in another state.
Answers 9–12
9. Losing. Only one counter can be removed, and taking the last is defined to lose in this variant. The old last-counter-wins classification cannot be reused. This one-counter terminal case is the foundation for the new pattern.
10. Remove three, leaving one. The opponent must take that last counter and lose. Removing all four is illegal anyway, since the maximum is three. Leaving a multiple of four is not the target here; the useful losing class is one more than a multiple of four.
11. Losing. The three legal moves leave four, three or two. From each, the opponent can leave one by removing three, two or one respectively. Every permitted choice therefore gives the opponent a winning continuation. A cooperative error could change the actual game, but not the perfect-play classification.
12. Remove one, leaving seventeen. Keep leaving counts congruent to one modulo four: seventeen, thirteen, nine, five and one. Complete the opponent’s removal to a total of four on each pair of turns. At one, the opponent takes the last and loses; no further response is required.
Answers 13–16
13. Add one first. The controlled totals are one, six, eleven, sixteen, twenty-one, twenty-six and thirty-one. After an opponent adds k, add five minus k. Every response is between one and four, and the final one reaches the exact target without overshooting.
14. Add one. The opponent has moved eleven to fifteen. Your addition of one reaches sixteen, maintaining a five-point increase over the complete pair of moves. If the opponent had added a different amount, your response would change so that the pair still totals five.
15. The second player. Thirty is a multiple of five remaining from the initial zero. The second can answer every first-player addition k with five minus k, reaching five, ten, fifteen, twenty, twenty-five and thirty. No first move breaks this legal response pattern.
16. Add two to win. Adding three or four would exceed thirty-one and is forbidden. Adding one remains legal but does not win immediately. The set of legal moves must be checked against the remaining distance, not copied unchanged from earlier turns with more room.
Answers 17–24
17. Mirror in the other pile. After the first player removes k from one pile, remove k from the other. Both piles are equal after every second-player response, and the response is legal because the other pile began the pair at the same size. Eventually the second player empties the final pile.
18. Remove three from the pile of eight. This leaves five and five for the opponent. Thereafter mirror each opponent removal in the other pile. Equalisation is legal because any positive amount up to the chosen pile’s size may be removed. The first player has taken the strategic mirroring role.
19. The untouched pile still has the original equal size. If the opponent can remove k from one of two equal piles, k cannot exceed that original size. The other pile therefore also contains at least k, making the matching response legal. The response restores equality and reduces the total.
20. Remove the other three; the second player wins. Only one pile may be touched per move, so the first cannot take all six at once. After they empty one pile, the remaining three can be taken legally from the other, making the second player the final-counter taker.
21. Remove three, leaving two. With two counters, the opponent can only remove one, leaving one for you to take and win. Taking four from five would leave one for the opponent, who would win immediately. The missing legal move two is what makes this game differ from the one-through-four version.
22. Losing. The legal successors are six, four and three. Six is winning by removing four to leave two; four and three are immediate wins by taking the pile. Since every legal move from seven gives the opponent a winning position, seven is losing under perfect play.
23. No. Five wins establish what happened in five sampled play histories. They do not cover every legal opponent reply. A proof needs a response rule that is legal and maintains the winning argument for every allowed reply, together with an explanation of why the game terminates in the desired result.
24. No. The table establishes classifications only for the checked range if its calculations are correct. An unrestricted claim needs a proof that the proposed pattern continues, such as a valid recurrence or residue argument covering all later cases. More checked entries are useful evidence but do not silently become an all-integers theorem.
10. Use the game to make reasoning observable
Ask the learner to explain what a move leaves for the opponent. “I take two because twenty-six leaves remainder two” is a start, but the next question is why leaving twenty-four helps. The learner should connect it to the opponent’s three legal choices and a legal response after each.
Keep the first explorations small. With four counters, all branches can be checked without a long calculation. With eight, the learner can reuse the established classification of five, six and seven. Build the strategy from explainable states rather than introducing a large starting pile and revealing a trick.
Separate a move from a strategy
One correct opening is not the full plan. After it, deliberately choose different legal replies across games and ask how the response changes. The learner should not need the opponent to follow a memorised script. A response rule such as “complete the pair to four” handles all three possibilities in one statement.
A fixed sequence of moves can fail when the opponent deviates. This is an important transferable distinction: planning under another agent’s choices requires contingent responses, not only a preferred future sequence.
Let a rule change test understanding
Switch from last-counter-wins to last-counter-loses and return to a pile of one. Ask which earlier conclusions change first. Then rebuild two, three, four and five. The child can see the new residue class arise from the ending rather than being asked to memorise a second unexplained list.
Next remove a legal option, such as allowing one, three and four but not two. Ask whether the old complementary response is still always available. A single missing response can invalidate the earlier proof. This is a precise reason to revise a strategy, not a vague instruction to “be flexible.”
An honest record of evidence
A lesson record can state the starting position, rule set, predicted move, actual opponent reply and explanation. Keep practice outcomes separate from mathematical claims. A winning position played badly may lose; a losing position may win against an error. Neither event alone refutes the correct perfect-play analysis.
Do not manufacture a tally of successful student trials or claim a strategy has been tested in live classrooms when only its mathematics has been checked. The worked analyses here concern the stated models, not observed learner performance.
Link games back to the wider mathematics
The multiples-of-four strategy applies remainder cycles. The two-pile mirror preserves an invariant. The winning/losing classification uses “some” and “every” from logic. The difference between a guaranteed strategy and a favourable random outcome connects to probability. These links let the game return to a mathematical purpose beyond winning a round.
A suggested final task is to change one rule and ask the learner to write a new small-state table, propose a pattern and test it. Require a distinction between what has been proved and what remains a conjecture. This is an adaptable teaching suggestion, not a guarantee of progress after a fixed number of games.
Return to the BTT Primary Mathematics Learning Hub for the prerequisite guides. A strong finishing sentence names the rules and the reason: “Under these moves and this ending, I can answer every legal reply and leave the opponent the stated class of positions.”
Sources and scope
The linked MOE page supplies school-scope context, while NRICH supplies related further exploration of target games. The numerical games, 24 practice questions, worked answers and proofs here are original exposition of explicitly stated mathematical models. No live game service, wagering, automated opponent or classroom experiment is part of this guide.